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On full cover property of ordered fields. (English) Zbl 0823.12006
The author defines the notion of a full cover of an interval $$[a,b]$$ in an ordered field $$A$$ as follows: A class $$S$$ of intervals $$I\subseteq [a,b]$$ is said to be a full cover of $$[a, b]$$ if $$\forall x\in [a,b]$$ $$\exists \delta (x)> 0_ A$$ in $$A$$ such that $$S$$ contains each closed interval $$I\subseteq [a,b]$$ of length less than $$\delta (x)$$ and containing $$x$$. Moreover, he defines that an ordered field $$A$$ has the full cover property (FC-property) provided that for each $$a,b\in A$$, $$a<b$$, the following statement holds: If $$S$$ is a full cover of $$[a,b]$$, then there exists a partition $$a_ 0= x_ 0< x_ 1< \cdots < x_ m= b$$, $$x_ j\in A$$, such that $$| x_{k-1}, x_ k|\in S$$, $$k=1,\dots, m$$. Then he proves that an Archimedean ordered field $$A$$ is complete iff it has the FC-property. He gives some applications of this property.
##### MSC:
 12J15 Ordered fields 12J10 Valued fields
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##### References:
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